Two Equals Four 


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Two Equals Four  2011/04/14
Consider the equation $2=x^{x^{x^{x^{\ldots}}}}$ and suppose we want to solve it for x. Because the exponential tower is infinite, we can also write it as $2=x^{\left({x^{x^{x^{\ldots}}}}\right)}$
But the part in brackets is the same as the whole, and hence is equal to 2. Thus we have $2=x^2.$ Hence $x=\sqrt{2}$ and so Now consider the equation and again let's solve for x. As before, we can write it as and again, the part in brackets is the same as the whole, and so now we get 4=x^{4} But take the square root of each side and we get 2=x^{2} and so again So now we have So is 2, and it's also 4. Hence 2=4 (and halving it means 1=2).
CommentsIn an email, Iain Murray has pointed to exercise 4.20 on page 86 of the book:

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